re: index, see here, re translations in English I don't believe they exist as RMF was opposed to them. Bits and pieces were translated in specific works, e.g., on his opinions re cardiac vs. brain death
– mblochApr 3 at 3:14

It is forbidden to publish abbreviations and rulings from responsa and to translate answers to the language of the state

... I heard that one of them had done a book in the Hebrew language in copying laws from the my sefer Igrot Moshe, which is prohibited even if the translation was appropriate [...] a number of people asked me for permission [to translate Igrot Moshe into English] and I told them I did not give permission for this.

Note: only translated small portions of the teshuva since it would be not only ironic but wrong to go against Rav Moshe's wishes in the exact teshuva where he writes he doesn't want Igrot Moshe to be translated

(hopefully the 2 snippets I translated were ok- in as much as they can help teach and serve as a disclaimer to what Rav Moshe's preferences in this matter are)

R. Feinstein's son in law R. Moshe Dovid Tendler published a translation/commentary of some of R. Feinstein's responsa relating to medical issues. The book is called Responsa of Rav Moshe Feinstein: Translation and Commentary Vol I Care of the Critically Ill. In his introduction R. Tendler discusses R. Feinsteins responsum (Y.D. 3:91) that prohibited the translation of his responsa, and R. Tendler cites a later responsum (Y.D. 4:38) to explain why his translation is okay. He also adds:

My great father-in-law, זצ"ל, in discussions at my home, where for decades he spent the Succos holy days, further elaborated on this point. He asked me to translate those responsa which I felt it was important for our brethren to understand, but to do so "as a rebbe teaches a talmid." He asked that I work with meticulous care, presenting all the relevant background material in a lucid style, with full awareness that the readership would span the spectrum from Torah scholars to people with minimal background. He added, as an aside, that an English translation of the responsa would be available to non-Jews as well. Although teaching Torah to non-Jews is prohibited, our purpose in translating these responsa is to instruct Jews, and therefore we need not consider this translation a violation of the edict concerning the study of Torah by non-Jews.

I have prepared the following translations in accordance with the above instructions. Although the English version is faithful to the original, I have included additional material, enclosed in brackets, to instruct and edify those who will turn to the writings of the greatest posek of our generation as to a personal rebbe.

Portions of this book are available online at Google Books, and it is available on Sefaria as well.

In Theology in the ResponsaR. Louis Jacobs goes through each century of responsa literature and summarizes those responsa that relate to theology. In the section for the twentieth century he summarizes three of R. Feinstein's responsa. However, as far as I know this cannot be read online.